Expected value conditioned on subset of domain

Question

I have a random variable $X$ with pdf $f(\cdot)$ and support $[0, b]$. As part of some results I am generating, I am left with the integral $$\int_a^b x f(x) \, dx$$ where $0<a<b.$ I am wondering how to think about this integral - can this be expressed $E[X\mid X>a]$ or something like this? Seems like a straightforward question I know but I couldn’t think of the right search terms to use. I’d welcome a link to another post asking the same thing if it exists.

Answer 1

Almost :) $$ \mathbb{E}[X\mid X>a] = \frac{1}{\mathbb{P}[X>a]} \int_a^b xf(x)\,dx = \frac{\int_a^b xf(x)\,dx}{\int_a^b f(x)\,dx} $$

Answer 2

If $S$ is any subset of $(a,b),$ then you have \begin{align} & \Pr(X\in S\mid X>a) = \frac{\Pr(X\in S\ \&\ X>a)}{\Pr(X>a)} \\[8pt] = {} & \frac{\Pr(X\in S)}{\Pr(X>a)} = \frac{\int_S f(x)\,dx}{\int_a^b f(x)\, dx}. \end{align} Therefore the conditional probability density of $X$ given that $X>a$ is $$ f_{X\,\mid\,X\,>\,a}(x) = \frac{f(x)}{\int_a^b f(u)\,du}. $$ So you integrate $x$ times that to get the conditional expected value.